Zadanie 2
Let F(x) = 3x4 − 4x3 + px2 + qx + 1. The polynomial has factor x + 1, and division by x − 2 leaves remainder 45. Determine p and q.
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Let F(x) = 3x4 − 4x3 + px2 + qx + 1. The polynomial has factor x + 1, and division by x − 2 leaves remainder 45. Determine p and q.
State V when (i) x = 0; (ii) x = a/3.
Show that V = 9x3 − 3(a + b)x2 + abx.
Show that the only solutions of dV/dx = 0 are .
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You may use that , that (a + b + D)/9 is outside [0, a/3], and that xm = (a + b − D)/9 is strictly inside it. Use (a) and (d) to explain why this local maximum is the largest capacity on the permitted interval.
For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b. Show that t > s.
For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b. You are given . Substitute into to show that xm = st/3.
Let xm denote the interior height maximizing V on [0, a/3]. For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b. Let Vm = xm(a − 3xm)(b − 3xm). (i) Explain why s divisible by 3 guarantees that both xm and…
A region is bounded above by y = 1 + e2x, below by y = 1 − 2e2x, and by the vertical lines x = 0 and x = ln 2. Find its exact area.